# Time constant

A **time constant** (symbol τ, the Greek letter tau) is the parameter that fixes the speed of a first-order exponential response: it is the time for the response to a step input to cover 63.2 percent of the distance to its final value, or equivalently for a decaying quantity to fall to 1/e (about 36.8 percent) of its initial value.<sup>[1](https://www.mindspaneducation.com/learn/control-systems/first-order-system-response/)</sup><sup> • </sup><sup>[2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)</sup> The same number appears in electrical circuits, thermal systems, chemical kinetics, and radioactive decay, because all of these are governed by first-order linear differential equations whose solutions are exponentials of the form \( e^{-t/\tau} \).<sup>[3](https://www.sciencedirect.com/topics/engineering/first-order-equation)</sup>

| Key fact | Detail |
|---|---|
| Defining value | After one time constant, a step response has reached 63.2% of its final change; a decaying quantity has fallen to 1/e ≈ 36.8% of its initial value<sup>[1](https://www.mindspaneducation.com/learn/control-systems/first-order-system-response/)</sup><sup> • </sup><sup>[2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)</sup> |
| RC circuit | \( \tau = RC \); units of ohms times farads are seconds<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)</sup> |
| RL circuit | \( \tau = L/R \); for a 7.50 mH inductor with a 3.00 Ω resistor, τ = 2.50 ms<sup>[4](https://openstax.org/books/college-physics-2e/pages/23-10-rl-circuits)</sup> |
| Response multiples | 63.2% at 1τ, 86.5% at 2τ, 95.0% at 3τ, 98.2% at 4τ, 99.3% at 5τ<sup>[3](https://www.sciencedirect.com/topics/engineering/first-order-equation)</sup> |
| Rise time | The 10%–90% rise time of a first-order RC response is approximately \( t_r \approx 2.2\tau \)<sup>[5](https://circuits.mit.edu/S26/psets/06/prelab)</sup> |
| Radioactive decay | The mean life τ = 1/λ is the time constant of the exponential \( e^{-t/\tau} \); the half-life is \( t_{1/2} = \ln 2 / \lambda \approx 0.693/\lambda \), so τ ≈ 1.4427 \( t_{1/2} \)<sup>[6](https://csrri.iit.edu/~howard/bootcamp/lechtml/radioac.html)</sup><sup> • </sup><sup>[7](https://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter1.pdf)</sup> |
| Practical settling | Five time constants is the commonly accepted interval for a capacitor to charge fully, reaching about 99.3% of the applied voltage<sup>[8](https://wiki.analog.com/university/courses/engineering_discovery/lab_2)</sup> |

## Definition and origin

The time constant arises from the first-order differential equation

\[ \tau \frac{dy}{dt} + y = K u, \]

whose transfer function is \( G(s) = \dfrac{K}{\tau s + 1} \).<sup>[9](https://www.youtube.com/watch?v=7IVzmKAwcb8)</sup> For a step input of amplitude A starting from zero initial conditions, the output is

\[ y(t) = KA\left(1 - e^{-t/\tau}\right). \]

Setting \( t = \tau \) gives \( 1 - e^{-1} = 0.632 \), which is where the 63.2 percent figure comes from.<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)</sup> The same constant governs decay: a quantity \( X(t) = X(0)\,e^{-t/\tau} \) has fallen to \( e^{-1} \) of its starting value after one time constant.<sup>[10](https://people.ohio.edu/piccard/radnotes/data.html)</sup>

The time constant also has a geometric meaning. At the instant the input starts, the output's initial rate of rise, extrapolated as a straight line, would complete the whole response in exactly one time constant; the actual exponential curve falls behind this tangent and approaches the final value asymptotically.<sup>[3](https://www.sciencedirect.com/topics/engineering/first-order-equation)</sup><sup> • </sup><sup>[11](https://kau.edu.sa/Files/0060757/Subjects/33%20Response%20of%20first%20order%20systems.pdf)</sup> For a process with gain \( K_p \), the initial rate of rise of the output is \( K_p/T \).<sup>[12](https://www.ect.ac.uk/resources/practical-process-control-for-engineers-and-technicians/)</sup>

## Response multiples

Because the response is exponential, the fraction completed after any multiple of τ is fixed, independent of the system. The standard table is:<sup>[3](https://www.sciencedirect.com/topics/engineering/first-order-equation)</sup>

| Time | % response | % dynamic error |
|---|---|---|
| 0 | 0.0 | 100.0 |
| 1τ | 63.2 | 36.8 |
| 2τ | 86.5 | 13.5 |
| 3τ | 95.0 | 5.0 |
| 4τ | 98.2 | 1.8 |
| 5τ | 99.3 | 0.7 |

Two settling-time conventions follow directly from this table. A first-order system settles within 5% of its final value after 3τ and within 2% after 4τ; control-engineering references treat the 4T (2% criterion) and 3T (5% criterion) numbers as distinct values, not interchangeable rules of thumb.<sup>[1](https://www.mindspaneducation.com/learn/control-systems/first-order-system-response/)</sup> In electronics practice, five time constants is the commonly accepted interval for a capacitor to charge fully.<sup>[8](https://wiki.analog.com/university/courses/engineering_discovery/lab_2)</sup>

**Rise time.** The rise time \( t_r \), defined as the time a signal takes to go from 10% to 90% of its final value, is related to the time constant by \( t_r \approx 2.2\tau \) for a first-order RC response; oscilloscopes measure rise time directly, so it provides a convenient route to τ in the laboratory.<sup>[5](https://circuits.mit.edu/S26/psets/06/prelab)</sup>

## Time constants in physical systems

**RC circuits.** For a resistor-capacitor circuit the time constant is \( \tau = RC \). During charging, the capacitor voltage follows \( V_C(t) = \epsilon\left(1 - e^{-t/\tau}\right) \) and the current decays as \( I(t) = I_0 e^{-t/\tau} \); during discharge the charge decays as \( q(t) = Q e^{-t/\tau} \).<sup>[2](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)</sup> A larger resistance or capacitance slows the transient; time constants are the same for charging and discharging.<sup>[8](https://wiki.analog.com/university/courses/engineering_discovery/lab_2)</sup> As a worked example, a 100 kΩ resistor with a 47 µF capacitor gives τ = 4.7 s, so full charge takes about 23.5 s.<sup>[8](https://wiki.analog.com/university/courses/engineering_discovery/lab_2)</sup>

**RL circuits.** For a resistor-inductor circuit the time constant is \( \tau = L/R \). When the supply is connected, the current rises as \( I = I_0(1 - e^{-t/\tau}) \); when it is removed, the current decays as \( I = I_0 e^{-t/\tau} \), falling to 0.368 of its initial value in each successive τ.<sup>[4](https://openstax.org/books/college-physics-2e/pages/23-10-rl-circuits)</sup> A larger inductance lengthens the transient because the inductor stores more magnetic energy and generates a stronger back-emf opposing the current change, while a larger resistance shortens it.<sup>[13](https://www.electronics-tutorials.ws/inductor/lr-circuits.html)</sup>

**Thermal systems.** A body cooling or heating by convection has a time constant that grows with density, volume, and specific heat, and shrinks with a larger heat transfer coefficient or surface area, matching the intuition that massive objects warm and cool slowly.<sup>[14](https://web.mit.edu/course/16/16.unified/www/FALL/thermodynamics/notes/node129.html)</sup>

**Process control.** A liquid tank with cross-sectional area A and outflow resistance R has time constant \( AR \). In one worked example, a tank of 2 m² area operating at a 2 m level with an outflow of 5 m³ per hour has R = 0.4 hr/m² and therefore τ = 0.8 hr, which predicts how long the level takes to stabilize after a small step change.<sup>[12](https://www.ect.ac.uk/resources/practical-process-control-for-engineers-and-technicians/)</sup>

**Radioactive decay.** [Radioactive decay](https://www.edgepedia.org/radioactive-decay) is a first-order process, \( N(t) = N_0 e^{-\lambda t} \). The reciprocal of the decay constant, \( \tau = 1/\lambda \), is called the mean life because it equals the average lifetime of the atoms in the population; the half-life is \( t_{1/2} = \ln 2/\lambda \approx 0.693/\lambda \), so the mean life is about 1.4427 times the half-life.<sup>[6](https://csrri.iit.edu/~howard/bootcamp/lechtml/radioac.html)</sup><sup> • </sup><sup>[7](https://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter1.pdf)</sup> The same exponential law describes first-order chemical reactions and the clearance of medications or toxins from blood by the liver.<sup>[10](https://people.ohio.edu/piccard/radnotes/data.html)</sup>

## Measuring and using the time constant

The 63.2% rule is the standard way to extract a time constant from a measured step response: record the steady-state value, find the time at which the output has covered 63.2% of the change, and read τ directly from the time axis.<sup>[15](https://cookierobotics.com/067/)</sup><sup> • </sup><sup>[16](https://engineering.purdue.edu/~byao/Courses/ME365%20-%20Yao%20-%20Fall%202016/L13_System%20Identification_Time-domain_Yao_filled.pdf)</sup>

Designers also place the time constant deliberately. In an RCD snubber that controls the rate of voltage rise at a MOSFET drain, the capacitor must fully charge and discharge each cycle, so the [RC time constant](https://www.edgepedia.org/rc-time-constant) is set to about one tenth of the switching period.<sup>[17](https://www.analog.com/en/resources/technical-articles/snubber-circuits-suppress-voltage-transient-spikes.html?gated=1757493644261)</sup> Conversely, when a square wave drives an [RC circuit](https://www.edgepedia.org/rc-circuit), the response reaches the input's full excursions only if the time constant is small compared with the half period; when the two are comparable, the output fails to reach the extremes and its ripple magnitude is inversely proportional to RC.<sup>[18](https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf)</sup>

## References

1. [Time Response of First-Order Systems, MindSpan Education](https://www.mindspaneducation.com/learn/control-systems/first-order-system-response/)
2. [10.5 RC Circuits, University Physics Volume 2, OpenStax](https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits)
3. [First-Order Equation, ScienceDirect Topics](https://www.sciencedirect.com/topics/engineering/first-order-equation)
4. [23.10 RL Circuits, College Physics 2e, OpenStax](https://openstax.org/books/college-physics-2e/pages/23-10-rl-circuits)
5. [Prelab: RC Circuits and Rise Time, 6.200 Spring 2026, MIT](https://circuits.mit.edu/S26/psets/06/prelab)
6. [Chapter 13: Radioactive Decay, mean life and half-life, IIT](https://csrri.iit.edu/~howard/bootcamp/lechtml/radioac.html)
7. [1.6 Radioactive Decay, MIT 12.201/12.501](https://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter1.pdf)
8. [Introduction to RC Circuits, Analog Devices Wiki](https://wiki.analog.com/university/courses/engineering_discovery/lab_2)
9. [First-order system response to step: gain, time constant (theory), Modeling, Identification, Control (UPV)](https://www.youtube.com/watch?v=7IVzmKAwcb8)
10. [Radiation Notes: The Analysis of Experimental Data, Ohio University](https://people.ohio.edu/piccard/radnotes/data.html)
11. [response of first order systems](https://kau.edu.sa/Files/0060757/Subjects/33%20Response%20of%20first%20order%20systems.pdf)
12. [Practical Process Control for Engineers and Technicians, UK ECT](https://www.ect.ac.uk/resources/practical-process-control-for-engineers-and-technicians/)
13. [Time Constant of an LR Series Circuit, Electronics Tutorials](https://www.electronics-tutorials.ws/inductor/lr-circuits.html)
14. [18.3 Transient Heat Transfer, MIT Unified Engineering](https://web.mit.edu/course/16/16.unified/www/FALL/thermodynamics/notes/node129.html)
15. [System Identification of 1st Order System, Cookie Robotics](https://cookierobotics.com/067/)
16. [System Identification in Time-Domain, ME365, Purdue University](https://engineering.purdue.edu/~byao/Courses/ME365%20-%20Yao%20-%20Fall%202016/L13_System%20Identification_Time-domain_Yao_filled.pdf)
17. [Snubber Circuits Suppress Voltage Transient Spikes, Analog Devices](https://www.analog.com/en/resources/technical-articles/snubber-circuits-suppress-voltage-transient-spikes.html?gated=1757493644261)
18. [Transient RC and RL Circuits, 6.071J, MIT OpenCourseWare](https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/3021be47ce51060507c5fa54373f2d78_trns1_rc_lc_rlc1.pdf)

---
*Topic: Encyclopedia › Physical world and mathematics › Physics › Classical physics › Electromagnetism › Electromagnetic quantities and history › Electromagnetic quantities › Impedance, resistance, and reactance quantities*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

*Copyright 2026 EdgeChat AI, a subsidiary of Biostate AI.*

License: Edgepedia Community License 1.0, https://www.edgepedia.org/edgepedia/license